Value of Tan 15

In trigonometry, the value of tan 15 is important, especially when solving problems involving triangles and angles. The value of tan 15 degrees is not found in the standard trigonometric table that includes common angles like 30°, 45°, or 60°. Instead, we use trigonometric identities to determine the value of tan 15 accurately. This article discusses various methods to derive and understand the value of tan 15, using basic trigonometric formulas and identities. We will also go through the steps to manually calculate the value of tan 15 degrees and use known values of sine and cosine to confirm the result.

 

Table of Contents

• What Is the Value of Tan 15?
• How to Find the Value of Tan 15 Using Sine and Cosine
• Importance of Knowing the Value of Tan 15 Degree
• Solved Example
• Trigonometric Table Reference
• Conclusion
• FAQs On Value of Tan 15

 

What Is the Value of Tan 15?

The value of tan 15 degrees can be found using the identity for the tangent of a difference of angles:  

tan(A - B) = (tan A - tan B) / (1 + tan A · tan B)  

To find tan 15, we set:  

tan(15°) = tan(45° - 30°)  

The standard values are:  

tan 45° = 1  

tan 30° = 1/√3  

Now we plug the values into the formula:  

tan(15°) = (1 - 1/√3) / (1 + 1 · 1/√3)  

Next, we multiply the numerator and denominator by √3 to rationalize:  

tan(15°) = (√3 - 1) / (√3 + 1)  

So, the value of tan 15 is:  

tan(15°) = (√3 - 1) / (√3 + 1)  

Now, approximating √3 ≈ 1.732:  

tan(15°) = (1.732 - 1) / (1.732 + 1) = 0.732 / 2.732 ≈ 0.2679  

Thus, the value of tan 15 degrees is approximately 0.2679.

 

How to Find the Value of Tan 15 Using Sine and Cosine

Another way to find the value of tan 15 is by using the identity:  

tan θ = sin θ / cos θ  

So,  

tan(15°) = sin(15°) / cos(15°)  

 

We can use these identities:  

sin(A - B) = sin A cos B - cos A sin B  

cos(A - B) = cos A cos B + sin A sin B  

Let’s use:  

sin(15°) = sin(45° - 30°)  

cos(15°) = cos(45° - 30°)  

We know these values:  

sin 45° = 1/√2  

cos 45° = 1/√2  

sin 30° = 1/2  

cos 30° = √3/2  

Now we apply the identities.

For the numerator (sin 15°):  

sin(15°) = (1/√2)(√3/2) - (1/√2)(1/2) = (√3 - 1) / (2√2)  

For the denominator (cos 15°):  

cos(15°) = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1) / (2√2)  

Now we divide sin 15° by cos 15°:  

tan(15°) = [(√3 - 1)/(2√2)] / [(√3 + 1)/(2√2)] = (√3 - 1)/(√3 + 1)  

So, the value of tan 15 is the same as before:  

tan(15°) = (√3 - 1)/(√3 + 1) ≈ 0.2679.

 

Why Is It Important to Know the Value of Tan 15?

The value of tan 15 degrees is often needed in solving real-world problems in physics, engineering, and architecture. When working with angles not listed in trigonometric tables, identities like angle subtraction help find the value of tan 15. This knowledge aids in simplifying expressions, calculating heights and distances, and working on trigonometric proofs.

 

Memorize These Key Points  

• value of tan 15 = (√3 - 1)/(√3 + 1)  

• value of tan 15 degrees ≈ 0.2679  

• This can be derived using: 
  tangent subtraction identity: tan(A - B) 
  sine and cosine identities  

 

Solved Example:

Example 1:
Evaluate the expression
(1 - tan 15°) / (1 + tan 15°)

Solution:
We know:
tan 15° = (√3 - 1) / (√3 + 1)

Substitute the value into the expression:

(1 - (√3 - 1)/(√3 + 1)) / (1 + (√3 - 1)/(√3 + 1))

Step 1: Simplify the numerator
= [(√3 + 1) - (√3 - 1)] / (√3 + 1)
= (√3 + 1 - √3 + 1) / (√3 + 1)
= 2 / (√3 + 1)

Step 2: Simplify the denominator
= [(√3 + 1) + (√3 - 1)] / (√3 + 1)
= (√3 + 1 + √3 - 1) / (√3 + 1)
= 2√3 / (√3 + 1)

Step 3: Combine numerator and denominator
= [2 / (√3 + 1)] ÷ [2√3 / (√3 + 1)]
= 2 / 2√3
= 1 / √3

Final Answer:
(1 - tan 15°) / (1 + tan 15°) = 1 / √3

 

Example 2:
Simplify:
tan 15° × tan 75°

Solution:
We know:
tan 15° = (√3 - 1)/(√3 + 1)
tan 75° = cot 15° = 1 / tan 15°

So,
tan 15° × tan 75° = tan 15° × (1 / tan 15°) = 1

Final Answer:
tan 15° × tan 75° = 1

 

Example 3:
Find the value of:
(tan 45° - tan 15°) / (1 + tan 45° × tan 15°)

Solution:
We know:
tan 45° = 1
tan 15° = (√3 - 1)/(√3 + 1)

Use the formula:
tan(A - B) = (tan A - tan B) / (1 + tan A × tan B)

So, the expression is:
tan(45° - 15°) = tan 30°
= 1/√3

Final Answer:
(tan 45° - tan 15°) / (1 + tan 45° × tan 15°) = 1/√3

 

Example 4:
Evaluate:
tan²(15°) + 1

Solution:
We know:
tan 15° ≈ 0.2679

So,
tan²(15°) + 1 = (0.2679)² + 1
= 0.0718 + 1
= 1.0718 (approx)

Final Answer:
tan²(15°) + 1 ≈ 1.0718

Trigonometric Table Reference

 

 

Use this table to find the value of tan 15 degree alongside other values.

 

Conclusion  

Understanding the value of tan 15 is vital for solving trigonometric problems involving uncommon angles. By learning both the angle subtraction formula and the sine-cosine method, students can confidently and accurately calculate the value of tan 15. Whether for tests, practical applications, or deeper learning, knowing how to compute tan 15 degrees enhances your problem-solving abilities in trigonometry. Keep practicing these identities to solidify your foundation in trigonometry. The value of tan 15 shows how mathematical identities make evaluating complex angles easier.  

 

Related Links

Trigonometric Identities – Learn essential trigonometric identities, including reciprocal, Pythagorean, and co-function identities, with proofs and examples for better understanding.

Trigonometry Formula - Discover all the important trigonometric formulas in one place for quick learning and easy exam revision.

Sin Cos Tan Values - Learn and memorize the standard sine, cosine, and tangent values for key angles to boost your trigonometry skills.

 

Frequently Asked Questions On Value of tan 15

1. What is the value of cot15 in trigonometry?

Ans: The value of cot 15° in trigonometry is approximately 3.732 or exactly:
cot(15°) = √3 + 2

 

2. What is the half angle formula for tan 15?

Ans: Tan 15° can be found using the angle subtraction formula:
tan(15°) = tan(45° − 30°) = (tan 45° − tan 30°) / (1 + tan 45° × tan 30°)
= (1 − √3/3) / (1 + 1 × √3/3)
= (3 − √3) / (3 + √3)

 

3. How to get tan 15?

Ans: To find tan 15°, use the identity:
tan(15°) = tan(45° − 30°) = (1 − √3/3) / (1 + √3/3)
This gives a value of approximately 0.2679.

 

4. What is the value of sin 15?

Ans: The value of sin 15° is approximately 0.2588 or exactly:
sin(15°) = sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°
= (1/√2 × √3/2) − (1/√2 × 1/2)
= (√3 − 1) / (2√2)

 

Master trigonometry concepts like tan 15° with Orchids The International School. Explore more math topics and boost your confidence today!